Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z=cos(x^2+y^2) that lies inside the cylinder x^2+y^2=1

If we substitute [tex]x=r\cos\theta[/tex] and [tex]y=r\sin\theta[/tex], we get [tex]r^2=x^2+y^2[/tex], so that[tex]z=\cos(x^2+y^2)=\cos(r^2)[/tex]which is independent of [tex]\theta[/tex], which in turn means the surface can be treated like a surface of revolution.Consider the function [tex]f(t)=\cos(t^2)[/tex] defined over [tex]0\le t\le1[/tex]. Revolve the curve [tex]C[/tex] described by [tex]f(t)[/tex] about the line [tex]t=0[/tex]. The area of the surface obtained in this way is then[tex]\displaystyle2\pi\int_C\mathrm dS=2\pi\int_0^1\sqrt{1+f'(t)^2}\,\mathrm dt[/tex][tex]=\displaystyle2\pi\int_0^1\sqrt{1+(-2t\sin(t^2))^2}\,\mathrm dt[/tex][tex]=\displaystyle2\pi\int_0^1\sqrt{1+4t^2\sin^2(t^2)}\,\mathrm dt\approx7.4144[/tex]